To compare the view vectors, estimated by different models, and compare our B-L strategy with traditional M-V portfolio using plug-in estimates of $\mu$ and $\Sigma$, we use two main evaluation metrics: cumulative returns and Sharpe ratio.  

\begin{figure*}[tpb]
\centering
\subfigure[Cumulative returns of different strategies.]{
	\includegraphics[width=7.5cm]{../results/eval_return.png}
	\label{fig:eval_return}
}
\subfigure[Sharpe ratio of different strategies.]{
	\includegraphics[width=7.5cm]{../results/eval_sharpe.png}
	\label{fig:eval_sharpe}
}
\label{fig:eval}
\end{figure*}

In the Figure. \ref{fig:eval}, we plot the cumulative returns \& sharpe ratio (over time) for the following strategies:
\begin{description}
\itemsep0em
\item[MV-Plugin-Expand]  Markowitz Model with plugin estimates using expanding window as training data;
\item[MV-Plugin-Roll]  Markowitz Model with plugin estimates using 5-year rolling window as training data;
\item[BL-ARMA-Expand]  Black-Litterman Model with ARMA view vector using expanding window as training data;
\item[BL-ARMA-Roll]  Black-Litterman Model with ARMA view vector using 5-year rolling window as training data;
\item[BL-AR-GARCH-Expand]  Black-Litterman Model with AR-GARCH view vector using expanding window as training data;
\item[BL-AR-GARCH-Roll]  Black-Litterman Model with AR-GARCH view vector using 5-year rolling window as training data;
\item[BL-AR-VIX-Expand]  Black-Litterman Model with AR-VIX view vector using expanding window as training data;
\item[BL-AR-VIX-Roll]  Black-Litterman Model with AR-VIX view vector using 5-year rolling window as training data;
\end{description} 

As we can see from Figure. \ref{fig:eval_return},  Black-Litterman approaches have better accumulative returns over time when compared the the Markowitz plug-in portfolio optimization model. By selecting training windows, the expanding window performs better than rolling window. However, this may change if we choose a larger rolling window. ARMA \& GARCH models perform better than VIX rolling window model. However, the model that has the largest cumulative returns and Sharpe ratio is the VIX expanding window predictive model.   Surprisingly, our Sharpe ratio goes down during time, as shown in Figure. \ref{fig:eval_sharpe}.